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Homework #4 - Continuous Random Variables and Probability Distributions

The fourth homework covers all of chapter 5.  The materials in those sections imply the following skills as being important: 

1. Identifying a random variable and the range of outcomes.
2. Identifying the probability density function (pdf) and cumulative density function (cdf) for a discrete random variable, and verifying that the pdf and/or cdf properties hold.
3. Manipulating the pdf and cdf to answer probability questions (e.g., P(X<5), P(X>=7.5), P(3<X<9)).
4. Identifying and recognizing random variables that conform to the definitions of the six "standard" random variables - normal, exponential, gamma, chi-square, and weibull. 
5. Manipulating the pdf and cdf of the standard discrete random variable formulations to determine probabilities associated with specific random variable ranges, random variable ranges that lead to specific probabilities.   

Assignment

Note about time:  The following assignment consists of essentially 8 questions (some questions refer to multiple problems in the text, but the different problems are continuations of one problem).  I estimate that each problem may take you approximately 30 minutes, resulting in a 4 hour assignment.  Because the time spent on the assignment is, itself, a random variable - the amount of time you spent may vary from this estimate.  If the assignment takes you significantly longer than 4 hours, please let us know via email.  Include in your email a statement of the amount of time the assignment required. 

1.  Mapping between Discrete and Continuous

Finish mapping between the uniform discrete and uniform continuous random variables.   The table used in lecture is available here.  Please print the table and complete the open items in the table. 

2.  Non-standard Continuous Probability Density Function

5-108, 5-109, 5-110 - All three of these problems refer to the same probability density function

3.  Exponential Distribution - Basic Calculations

5-69

4.  Exponential Distribution - in Practice

5-111, 5-112, 5-113 - All three of these problems refer to the same random variable and probability density function.

5.  Normal Distribution - Using the Tables

5-40 and 5-41:  Both problems refer to the same random variable

6.  Normal Distribution - in Practice

5-117 and 5-118:  Both problems refer to the same random variable

7.  Normal Approximations for Discrete Distributions

5-59

8.  Identifying Random Variables in Your Discipline:  The purpose of this problem is to encourage you to think about what constitutes a  random variable in your discipline.

(a). State your disipline (major)

For each of the next parts of the problem, you should (1) identify a discipline-relevant random variable with the properties requested and (2) describe the range of the random variable:

(b).     1 normally distributed random variable
(c).     1 exponentially distributed random variable
(d).     1 weibull distributed random variable.  
(e).     1 gamma distributed random variable

Hint:  To gain guidance on this problem, you might look to the examples and problems in sections 5-5 through 5-8 for ideas about identifying random variables for each of the "standard" continous distributions.  Finally, in determining random variables for the exponential and gamma distributions, you might consider the relationships among the definitions of the random variables for these distributions.

Questions and Answers

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Solutions

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